Theorem dmqmap

Description: QMap preserves the domain. Confirms that QMap is defined exactly on the points where cosets [ x ] R make sense (those in dom R ). (Contributed by Peter Mazsa, 14-Feb-2026)

	Ref	Expression
Assertion	dmqmap	⊢ ( 𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		df-qmap	⊢ QMap 𝑅 = ( 𝑥 ∈ dom 𝑅 ↦ [ 𝑥 ] 𝑅 )
2		ecexg	⊢ ( 𝑅 ∈ 𝑉 → [ 𝑥 ] 𝑅 ∈ V )
3	2	adantr	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑥 ∈ dom 𝑅 ) → [ 𝑥 ] 𝑅 ∈ V )
4	1 3	dmmptd	⊢ ( 𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅 )